Explore advanced concepts in general relativity through this lecture delivered by Zoe Wyatt from the University of Cambridge as part of the Fields Institute's General Relativity Winter School. Delve into sophisticated mathematical frameworks and theoretical foundations that underpin Einstein's theory of general relativity, building upon previous lectures in the series. Examine the geometric interpretation of spacetime, tensor calculus applications, and the mathematical tools essential for understanding gravitational phenomena. Learn how general relativity connects to broader themes in nonlinear evolution equations and their applications in physical sciences. Gain insights into current research directions and mathematical techniques used in modern gravitational physics, presented within the context of the Fields Institute's thematic program on shocks and singularities in nonlinear evolution equations.